Development of a Clinical Type 1 Diabetes Metabolic System Model and in Silico Simulation Tool

Abstract

Objectives

The goal of this study was to develop a system model of type 1 diabetes for the purpose of in silico simulation for the prediction of long-term glycemic control outcomes.

Methods

The system model was created and identified on a physiological cohort of virtual type 1 diabetes patients (n = 40). Integral-based identification was used to develop (n = 40) insulin sensitivity profiles.

Results

The n = 40 insulin sensitivity profiles provide a driving input for virtual patient trials using the models developed. The identified models have a median (90% range) absolute percentage error of 1.33% (0.08–7.20%). The median (90% range) absolute error was 0.12 mmol/liter (0.01–0.56 mmol/liter). The model and integral-based identification of SI captured all patient dynamics with low error, which would lead to more physiological behavior simulation.

Conclusions

A simulation tool incorporating n = 40 virtual patient data sets to predict long-term glycemic control outcomes from clinical interventions was developed based on a physiological type 1 diabetes metabolic system model. The overall goal is to utilize this model and insulin sensitivity profiles to develop and optimize self-monitoring blood glucose and multiple daily injection therapy.

Introduction

The control of type 1 diabetes is a widely studied and experimented research field. Previously published control methods are diverse, using different routes of insulin administration and glucose measurement. Since the 1970s, the closed-loop artificial endocrine pancreas has been heralded as the solution (as reviewed in Bequette¹). While no commercial product currently exists, the systems in current clinical use that are likely to constitute the components of an extracorporeal artificial pancreas are the continuous subcutaneous insulin infusion (CSII) pump and a continuous glucose measurement (CGM) device. Advanced control algorithms and methods to “close the loop” have also been widely studied (as reviewed elsewhere^2–4), despite early and ongoing limitations in sensors and pumps. Currently, the use of open-loop CGM and/or CSII has resulted in, at best, a modest clinical advantage over conventional methods of insulin administration or multiple daily injection (MDI) (as reviewed elsewhere^5,6). Additionally, these systems are only used by a small population of type 1 diabetic patients because of high upfront costs, costs of consumables, complexity, and the extensive health care infrastructure and support required. The prevalence of CSII use is as low as 2% of the type 1 diabetes population in the United Kingdom and up to 15–20% elsewhere and in the United States.⁷

Hence, there is a more practical and urgent need to address the large majority of the type 1 diabetes population using conventional glucose measurement, i.e., self-monitoring blood glucose (SMBG), and insulin administration, i.e., MDI methods, and for whom current conventional or intensive therapies are failing to deliver recommended levels of glycemic control.⁸ In the United States, over 50% of diagnosed diabetics aged 20–64 are deemed “out of control.⁹” The higher accuracy of bedside capillary blood glucose meters^10,11 and the latest insulin analogues for MDI therapy,¹² coupled with better control methods, have the potential to provide better care to the majority of outpatient or ambulatory type 1 diabetics than currently observed. Such techniques must necessarily be simple to implement to ensure broad clinical uptake by the diabetes population.

This study reports the development of a system model of the type 1 insulin–glucose regulatory system and its identification on a virtual patient cohort. The models utilized have several novel and unique features. In particular, the insulin model used is unique and captures the insulin kinetics of multiple insulin types in a single pharmacokinetic model for all shared physiological spaces. The pharmacodynamic model used has not been reported previously but bears components of a similar nature to other such models used in this field due to the need to capture similar physiology. This study is the basis for a novel model-based application to develop a simple and practical adaptive method for clinical glycemic control of type 1 diabetes using multiple daily injection and self-monitoring blood glucose measurements. In addition, the modeling of long-term clinical outcomes of glycemic control and their corroboration against clinical expectations and studies are explored further in a subsequent in silico simulation on a virtual patient cohort, which is also reported in this issue. Later, the complex interaction of all quantifiable errors in protocol application is investigated in a Monte Carlo study to test the robustness of the developed protocol in effectiveness and safety.¹³

Model Development

The system model shown in Equation (1) is an evolution of the model of Chase et al.¹⁴ and Wong et al.¹⁵

$$\dot{G}\left(t\right)=EG{P}_{0-G}-p_{G}G\left(t\right)-S_{1}G\left(t\right)Q\left(t\right)-RGC\left(t\right)-CNS+P\left(t\right)$$ (1)

where G(t) is plasma glucose concentration (mmol/liter), CNS is central nervous system glucose uptake (mmol/liter·min), EGP_0-G is endogenous glucose production extrapolated to zero plasma glucose concentration (mmol/liter·min), p_G is glucose effectiveness (min⁻¹), S_I is insulin sensitivity (liter/min·mU), Q(t) is interstitial (effective) insulin concentration (mU/liter), RGC(t) is renal glucose clearance (mmol/liter·min), and P(t) is meal plasma glucose rate of appearance (mmol/liter·min).

This glucose model differs mathematically from the model developed by Chase et al.¹⁴ and Wong et al.¹⁵ in removal of insulin effect saturation and addition of the renal glucose clearance rate, RGC(t). These two studies were done on highly dynamic, critically ill patients with high effective insulin resistance who were treated with intravenous insulin doses. Removal of the insulin effect saturation was deemed suitable for modeling more compliant, insulin-sensitive and stable type 1 diabetes patients treated with subcutaneously administered insulin.

$$RGC\left(t\right)=\left\{\begin{array}{ll}\frac{GFR}{V_p{m}_b}\left(G\left(t\right)-RGT\right)\hfill & \text{if\hspace{0.17em}}G\left(t\right)>RGT\hfill \\ 0\hfill & \text{if\hspace{0.17em}}G\left(t\right)\le RGT\hfill \end{array}\right\}$$ (2)

where RGC(t) is renal glucose clearance (mmol/liter·min), GFR is glomerular filtration rate (liter/min), G(t) is plasma glucose concentration (mmol/liter), RGT is renal glucose threshold (mmol/liter), V_p is glucose distribution volume (liter/kg), and m_b is body mass (kg).

Referring to Equation (2), the renal glucose clearance rate, RGC(t), models glucose removal by the kidney above the renal glucose threshold, RGT, using a linear relationship proportional to the glucose concentration above RGT and the glomerular filtration rate, GFR. From the study by Johansen and colleagues,¹⁶ this linear approximation is acceptable. Linear models have also been used in AIDA¹⁷ by Lehmann et al.¹⁸ and by Arleth et al.¹⁹

Insulin absorption from subcutaneous injection or infusion has been widely studied since Binder.²⁰ A novel, compartmental model of subcutaneous insulin absorption kinetics specifically developed for diabetes decision support has been presented.^21,22 The model accounts for the volume and concentration dependence of regular human insulin absorption and models the absorption kinetics of six insulin types, including monomeric insulin and insulin glargine. Additionally, insulin injected or infused subcutaneously or intravenously can also be modeled. A schematic of the model adapted from Wong and colleagues²² is shown in Figure 1. This model is used in this study, which is the first application of the developed model in the control of type 1 diabetes.

Figure 1.

Structure of the subcutaneous insulin absorption kinetic model. The model is characterized by a common hexameric state compartment for regular insulin (RI), neutral protamine Hagedorn (NPH) insulin, and lente insulin (x_h), whereas those for insulins glargine and ultralente (x_h,ulen and x_h,gla) are separate. A crystalline state compartment for NPH (c_NPH), lente (c_len), and ultralente (c_ulen) insulins and a precipitate compartment for insulin glargine (p_gla) model these protraction mechanisms. All insulin flows through a common a dimeric–monomeric state compartment (x_dm), interstitium compartment (x_i), and finally into the plasma (I). Adapted from Wong et al.²²

Modeling of the meal glucose rate of appearance (R_a), including digestion, absorption, and transport of glucose, is a complex process not widely studied.²³ Meal carbohydrate amount and type are the main factors affecting meal glucose R_a or P(t) in Equations (1) and (6).^24,25 However, clinical models of glucose R_a almost universally accept input of meal glucose amount only.^18,26 Glucose equivalent carbohydrate introduced by Yates and Fletcher²³ to express carbohydrate values as monosaccharide equivalents necessarily depends on an a priori known content of the carbohydrate type within the meal to be consumed, which is typically unavailable. Carbohydrate counting is a technique^27–29 commonly taught by diabetes care providers to improve glycemic management. Glycemic index, a measure of the effect of carbohydrate type, is not easily calculable for mixed meals³⁰ nor as readily available as carbohydrate content. The minimal models of meal glucose R_a by Worthington³¹ and Lehmann et al.^17,18 form the basis of the model used in this study. Referring to Figure 2 and Equations (3)–(5), the model consists of two compartments for the stomach and gut, with linear gastric emptying and gut-absorption rates to describe the plasma glucose R_a in Equation (6).Another simplification is the expression of meal carbo-hydrate content (in grams) as equivalent to the same mass of glucose monosaccharide regardless of the meal carbohydrate type.²³ Again, such meal data are typically unavailable in a clinical setting. As such, the complex digestion processes, such as the hydrolysis of poly-saccharides, are assumed linear and lumped into the simplified processes given earlier.

$$\frac{\text{dSTO}\left(T\right)}{\text{d}t}=-k_6\text{STO}\left(t\right)+u_{CHO}\left(t\right)$$ (3)

$$\frac{\text{dGUT}\left(t\right)}{\text{d}t}=\text{GABS}\left(t\right)+k_6\text{STO}\left(t\right)$$ (4)

$$\text{GABS}\left(t\right)=-\text{min}\left(k_7\text{GUT}\left(t\right),{\text{GABS}}_{\text{max}}\right)$$ (5)

$$P\left(t\right)=\frac{\text{GABS}\left(t\right)}{0\text{.18}\left(V_p{m}_b\right)}$$ (6)

where STO(t) is mass of carbohydrate/glucose in the stomach (g), GUT(t) is mass of carbohydrate/glucose in the gut (g), GABS is gut carbohydrate/glucose absorption rate (g/min), GABS_max is maximum gut carbohydrate/glucose absorption rate (g/min), k₆ is carbohydrate/glucose gastric emptying rate (min⁻¹), k₇ is carbohydrate/glucose gut-absorption rate (min⁻¹), u_CHO(t) is meal carbohydrate/glucose input (g/min), P(t) is meal plasma glucose rate of appearance (mmol/liter·min), V_p is glucose plasma distribution volume (liter/kg), and m_b is body mass (kg).

Figure 2.

Structure of the meal glucose rate of the appearance model. The model is characterized by a delta function to describe meal glucose input [u_CHO(t)], linear gastric emptying (k₆), and gut absorption (k₇) rates and saturable gut absorption (GABS_max).

Worthington³¹ found that the one-compartment model with time delay had the smallest fitting error. However, this result was obtained with a model fit to plasma glucose data and is dependent on the model of glucose kinetics used. Lehmann and colleagues¹⁸ used a “complex” function to describe the gastric emptying rate from the stomach compartment. This study used a linear transport rate, k₆, while the glucose input into the stomach compartment, u_CHO(t), is described by a delta function. Similar to the saturable gastric emptying rate of Lehmann et al.,¹⁸ this study incorporated a saturable gut-absorption rate, GABS_max. Saturable gut absorption has been postulated by Korach-Andre and colleagues³² in experiments using relatively large starch meals. However, this difference is likely to be small considering the minimal nature of both models. Referring to Figure 3, the effective gut-absorption rate is shown as a function of the mass of carbohydrate/glucose in the GUT compartment. Addition of the saturable term GABS_max effectively makes the gut absorption rate nonlinear as a function of the amount of carbohydrate in the gut. This dynamic is similar to that of the nonlinear three-compartment model of Dalla Man et al.,²⁶ which consists of dual stomach compartments with a nonlinear gastric emptying rate with four identified parameters. The nonlinear gastric emptying term is described by a hyperbolic tangent function as a function of the proportion of the consumed carbohydrate remaining in the stomach. There is no saturation term considered for large, absolute meals.

Figure 3.

Qualitative plot of the effective gut absorption rate as a function of the mass of carbohydrate/glucose in the GUT compartment. While the process of gastric emptying is linear, addition of the saturable gut absorption term, GABS_max, of 1.1 g/min effectively makes the process of gut absorption, and hence meal glucose R_a, nonlinear. At low glucose levels in the gut, the effective gut absorption rate is 0.0097 liter/min.

Referring to Table 1, the values of patient-independent model population constants are a priori identified from literature. The renal glucose threshold, RGT, has been shown to vary considerably in type 1 diabetes,¹⁶ but median values of 10 mmol/liter have been widely reported. The glucose distribution volume, V_p, is taken to be 0.22 liter/kg, the same value used by Lehmann et al.¹⁸ The glomerular filtration rate, GFR, is taken as 0.12 liter/min or 120 ml/min, which reflects the average adult GFR of 125 ml/min.³³

In a study by Dalla Man et al.,³⁴ the maximum meal R_a (R_{a meal}) was ∼8–9 mg/kg·min after an oral dose of 1 g/kg glucose. In the study by Korach-Andre et al.,³² the exogenous meal R_a (R_{a exo}) was approximately 7–9 mg/kg·min for a meal of 4 g/kg of starch (∼4.4 g/kg glucose). Despite the fourfold increase in glucose load, the maximum R_a remains at ∼9 mg/kg·min or ∼0.72 g/min for an average adult. In a study by Noah et al.,³⁵ a higher figure still of 11 mg/kg·min was reported in a porcine model. The maximum value of the rate of gut absorption is taken as 1.1 g/min using the Noah et al.³⁵ study as a basis, assuming a 100-kg body weight.

The proportion of glucose lost to first-pass splanchic uptake is still being debated, with proportions from negligible^36,37 to as high as 30% reported in some studies.³⁸ As there will be no tracer data in the intended application of the model, negligible losses from first-pass splanchnic sequestration and complete absorption are assumed for simplicity³⁹ with complete absorption of meal glucose. The values of k₆ and k₇ are optimized using nonlinear least squares to model-independent, mixed-meal tracer glucose R_a data³⁴ (results not shown).

The values of CNS, EGP_0-G, and p_G are derived from results of studies done by Del Prato et al.^40,41 Like the minimal model of Bergman and colleagues,⁴² the model is unable to differentiate noninsulin-mediated glucose uptake from production, which is lumped into a linear relationship with glucose. Referring to Figures 4 and 5, total body glucose uptake (TBGU) and hepatic glucose production (HGP) data described elsewhere^40,41 are used to identify CNS, EGP_0-G, and p_G. Data at glucose exceeding the approximate RGT of 10 mmol/liter are ignored to eliminate the need to evaluate renal glucose clearance, RGC, and associated errors. Under fasting and insulinopenic conditions, the P(t) and S_IG(t)Q(t) terms of Equation (1) can be further eliminated. By the linear definition of the effect of hyperglycemia on TBGU, CNS can then be derived as the “virtual” y intercept of the linear TBGU curve. The term “virtual” is used, as no glucose uptake is theoretically possible at zero glucose. The central nervous system glucose uptake CNS is saturated at 3.3 mmol/liter and is relatively insensitive to insulin and glucose.^43,44 At euglycemia, CNS accounts for ∼70% of all noninsulin-mediated glucose uptake,⁴⁵ and this proportion is likely to increase with hypoglycemia. Hence, use of the term CNS for the virtual y intercept of the linear TBGU curve is justified.

Figure 4.

Using HGP and TBGU data from Del Prato and colleagues⁴¹ for an insulinopenic normal cohort, values of CNS = 1.4 mg/kg·min, EGP_0-G = 2.6 mg/kg·min, and p_G = 0.006 min⁻¹ can be calculated by linear regression.

Figure 5.

Using HGP and TBGU data of Del Prato and colleagues⁴⁰ for an IDDM cohort under basal insulin, values of CNS = 1.7 mg/kg·min, EGP_0-G = 3.0 mg/kg·min, and p_G = 0.006 min⁻¹ can be calculated by linear regression.

Similarly, by the linear definition of the effect of hyperglycemia on HGP, EGP_0-G is the y intercept of the linear HGP curve and p_G is the slope of the combined TBGU and HGP curve. Hence, p_G is similar to the minimal model glucose effectiveness, S_G, but is defined under conditions of insulinopenia or subbasal insulin rather than basal insulin.⁴⁶

Unlike the minimal model, the insulin model in this study models the absolute insulin concentration, not insulin concentration above basal. In type 1 diabetes, conditions of basal insulin are not necessarily met all the time. Using data from Del Prato and colleagues⁴¹ for an insulinopenic normal cohort (Figure 4), values of CNS = 1.4 mg/kg·min, EGP_0-G = 2.6 mg/kg·min, and p_G = 0.006 min⁻¹ are obtained compared to CNS = 1.3 mg/kg·min, EGP_0-G = 3.0 mg/kg·min, and p_G = 0.009 min⁻¹ under basal insulin conditions (figure not shown). Compared to insulinopenia, the presence of basal insulin results in overestimation of p_G, although this value is still approximately half that of published values of the minimal model S_G for a normal cohort ∼0.024 min⁻¹.⁴⁶

Using data of Del Prato et al.⁴⁰ for an insulin-dependent diabetes mellitus (IDDM) cohort under basal insulin (Figure 5), values of CNS = 1.7 mg/kg·min, EGP_0-G = 3.0 mg/kg·min, and p_G = 0.006 min⁻¹ were obtained. Hence, p_G of the normal, insulinopenic cohort⁴¹ is similar to the IDDM cohort under basal insulin.⁴⁰ This result is logical, as S_G is decreased in IDDM,⁴⁶ whereas basal insulin increases S_G, either by increased glucose uptake⁴⁴ or by suppression of endogenous glucose production.⁴¹ In IDDM, the p_G obtained is also approximately half that of published S_G values of ∼0.013 min⁻¹.⁴⁶ One explanation is the elimination of data at high glucose concentrations from the p_G analysis, which, if unaccounted for, would include the effect of urinary glucose excretion, thereby increasing the “effective” glucose uptake. From this investigation, it can be deduced that for an IDDM cohort under conditions of insulinopenia, p_G must have an upper bound of 0.006 min⁻¹, which is assumed in this study. The values of CNS obtained are in agreement with other results,^45,47,48 and the assumption that CNS is approximately equal to the virtual y intercept of the linear TBGU curve is valid. A summary of the values of the model constants is shown in Table 1.

Table 1.

A Priori Identified Model Constants Obtained from the Literature Except Linear Gastric Emptying and Gut Absorption Rates (k₆ and k₇, Respectively), Which Were Optimized Using Nonlinear Least Squares to Model-Independent, Mixed-Meal Tracer Glucose R_a Data³⁴

Model constants	Values (units)
GABSmax	1.1 (g/min)
pG	0.0060 (min−1)
CNS	1.7 (mg/kg·min)
EGP0-G	3.0 (mg/kg·min)
GFR	0.12 (liter/min)
RGT	10 (mmol/liter)
VP	0.22 (liter/kg)
k6	0.0388 (min−1)
k7	0.0097 (min−1)

Methods

Patient Cohort

Patient data used in this study were obtained from AIDA on-line², the Web-based version of the AIDA educational diabetes program.⁴⁹ AIDA on-line² incorporates the physiological model developed by Lehmann et al.¹⁸ and can simulate glycemic levels for any insulin or meal stimuli over a period of 1 day. Patient data (n = 40) for this study were obtained from sample diabetes case scenarios available with AIDA on-line². Referring to Table 2, each patient case is unique in body weight, meals/carbohydrates consumed, and insulin treatment. Each patient also has unique clinical variables of hepatic and peripheral insulin sensitivity, glucose renal threshold, and glomerular filtration rate. Hence, the AIDA on-line² cohort represents a broad range of patients and possible clinical behavior. To retrieve blood glucose, plasma insulin, and meal glucose absorption rates from AIDA on-line², the “Advanced” display is selected to output data in text format. A sample of these data is shown in Figure 6.

Table 2.

Details of Patient Cohort (n = 40) from AIDA² On-line Showing Body Weight, Total Carbohydrate Consumed, Total Prandial Insulin Dose, Total Basal Insulin Dose, and Unique Clinical Variables of Hepatic and Peripheral Insulin Sensitivity, Glucose Renal Threshold, and Glomerular Filtration Rate

Case number	Body weight (kg)	Total carbohydrate consumed (g)	Total prandial insulin dose (U)	Total basal insulin dose (U)	Renal threshold	Renal function	Hepatic insulin sensitivity	Peripheral insulin sensitivity
1	70	130	7	30	Normal	Normal	Reduced	Reduced
2	68	180	13	10	High	Normal	Increased	Increased
3	70	120	9	13	Normal	Normal	Normal	Increased
4	60	180	13	12	Normal	Normal	Increased	Increased
5	98	180	12	12	Normal	Normal	Normal	Normal
6	76	120	8	28	Normal	Normal	Reduced	Increased
7	70	90	7	24	Normal	Normal	Increased	Reduced
8	70	120	10	20	Normal	Normal	Reduced	Increased
9	70	180	12	12	High	Normal	Normal	Increased
10	70	120	15	8	Normal	Normal	Normal	Increased
11	70	205	16	22	Normal	Normal	Normal	Increased
12	70	185	24	20	Normal	Normal	Reduced	Increased
13	76	100	8	26	Normal	Normal	Normal	Increased
14	65	70	5	20	Normal	Normal	Reduced	Normal
15	99	115	6	42	Normal	Normal	Reduced	Normal
16	70	180	9	32	Normal	Normal	Reduced	Increased
17	70	110	10	24	Normal	Normal	Reduced	Increased
18	60	165	18	36	High	Normal	Reduced	Increased
19	60	180	12	12	Normal	Normal	Normal	Increased
20	70	105	8	36	Normal	Normal	Normal	Increased
21	98	295	34	40	Normal	Normal	Reduced	Reduced
22	75	70	Biphasic 40		Normal	Normal	Reduced	Reduced
23	87	177	38	12	Normal	Normal	Reduced	Increased
24	76	95	7	24	Normal	Normal	Normal	Increased
25	70	120	18	28	Normal	Normal	Reduced	Normal
26	80	170	Biphasic 20		Normal	Normal	Normal	Increased
27	70	120	9	13	Low	Normal	Normal	Normal
28	75	85	5	40	Normal	Normal	Normal	Increased
29	83	60	12	25	Normal	Normal	Reduced	Normal
30	80	165	16	36	Normal	Normal	Reduced	Increased
31	99	220	29	14	Low	Normal	Normal	Increased
32	90	70	0	28	Normal	Normal	Reduced	Reduced
33	98	180	0	18	Normal	Normal	Normal	Increased
34	60	175	17	13	High	Normal	Normal	Increased
35	60	170	22	10	Normal	Normal	Normal	Increased
36	70	100	8	32	Normal	Normal	Reduced	Increased
37	70	105	9	36	High	Normal	Reduced	Reduced
38	70	95	Biphasic 26		Normal	Normal	Normal	Reduced
39	70	110	12	35	High	Normal	Reduced	Increased
40	76	100	7	30	Normal	Normal	Normal	Normal

Figure 6.

Sample raw blood glucose, plasma insulin level, and glucose absorption rate data from AIDA on-line².⁴⁹

Simulation Method

For in silico simulation, the virtual patient method is used.^50,51 This method has been utilized to develop effective glycemic control protocols by simulating the physiological glycemic response to glucose and insulin stimuli.^50–52 The glycemic responses are generated with patient-specific S_I(t) profiles derived from retrospective data. This clinically validated method⁵⁰ enables extensive simulations to be performed in a short time for rapid development and testing of any control methodology. In silico simulation was performed using MATLAB^® (The Mathworks, Natick, MA) implemented on a personal computer notebook (Pentium M 1.7 GHz).

To obtain the retrospective S_I(t) patient data profiles, the model is first fitted to data using the linear and convex, integral-based parameter identification method.⁵³ Equation (1) can be expressed in a generic integral form [Equation (7)] for period t_i-1 to t_i, which can then be rearranged into a set of linear equations as shown in Equation (9). All quantities in Equation (7) are modeled and, as such, are known except for G(t).

$$\begin{array}{l}{\displaystyle {\int }_{t_{i-1}}^{t_i}\dot{G}\left(t\right)\text{d}t={\displaystyle {\int }_{t_{i-1}}^{t_i}\left[EG{P}_{0-G}-p_{G}G\left(t\right)-{\overline{S}}_{I,i}G\left(t\right)Q\left(t\right)-RGC\left(t\right)-CNS+P\left(t\right)\right]\text{d}t}}\hfill \\ G\left(t_i\right)-G\left(t_{i-1}\right)={\displaystyle {\int }_{t_{i-1}}^{t_i}\left[EG{P}_{0-G}-RGC\left(t\right)-CNS+P\left(t\right)\right]\text{d}t}\hfill \\ -p_G{\displaystyle {\int }_{t_{i-1}}^{t_i}G\left(t\right)\text{d}t}\hfill \\ -{\overline{S}}_{I,i}{\displaystyle {\int }_{t_{i-1}}^{t_i}G\left(t\right)Q\left(t\right)\text{d}t}\hfill \\ {\overline{S}}_{I,i}{\displaystyle {\int }_{t_{i-1}}^{t_i}G\left(t\right)Q\left(t\right)\text{d}t={\displaystyle {\int }_{t_{i-1}}^{t_i}\left[EG{P}_{0-G}-RGC\left(t\right)-CNS+P\left(t\right)\right]\text{d}t}}\hfill \\ -p_G{\displaystyle {\int }_{t_{i-1}}^{t_i}G\left(t\right)\text{d}t}\hfill \\ -\left[G\left(t_i\right)-G\left(t_{i-1}\right)\right]\hfill \end{array}$$ (7)

$$G\left(t\right)={\displaystyle \sum _{i=1}^N\left[{\overline{G}}_{i-1}+\left({\overline{G}}_i-{\overline{G}}_{i-1}\right)\left(\frac{t-t_{i-1}}{t_i-t_{i-1}}\right)\right]}\left(H\left(t-t_{i-1}\right)-H\left(t-t_i\right)\right)$$ (8)

AIDA on-line² uses a first-order Euler integration method with a 15-minute step size to solve the plasma glucose model equation.⁵⁴ To determine G(t) to solve Equation (7), AIDA on-line² glucose data are interpolated linearly to obtain a piecewise linear G(t) function [Equation (8)].

The t_i – t_i-1 time interval for the optimization of S_I(t) is chosen arbitrarily as 10 minutes. Equation (8) is solved using a proprietary MATLAB linear solver. Referring to Figure 7 and Equations (9) and (10), a stepwise, time-variant S_I(t) with a 10-minute step interval is obtained.

$$\overline{A}\left\{{\overline{S}}_{I,i}\right\}=b$$ (9)

where

$$\begin{array}{l}\overline{A}={\displaystyle {\int }_{t_{i-1}}^{t_i}G\left(t\right)Q\left(t\right)\text{d}t}\hfill \\ \overline{b}={\displaystyle {\int }_{t_{i-1}}^{t_i}\left[EG{P}_{0-G}-RGC\left(t\right)-CNS+P\left(t\right)\right]dt}\hfill \\ -p_G{\displaystyle {\int }_{t_{i-1}}^{t_i}G\left(t\right)\text{d}t}\hfill \\ -\left[G\left(t_i\right)-G\left(t_{i-1}\right)\right]\hfill \end{array}$$

$$S_I={\displaystyle \sum _{i=1}^N{\overline{S}}_{I,i}\left(H\left(t-t_{i-1}\right)-H\left(t-t_i\right)\right)}$$ (10)

Figure 7.

Sample patient S_I(t) profile as obtained from model fit. Note the 10-minute interval for fitting the stepwise, time-variant S_I(t).

A proprietary MATLAB numerical ODE solver is used to solve the model equations with a 1-minute time step. Biphasic insulin preparations are treated as in AIDA with the insulin response assumed to be a superposition of the individual components of the preparation.¹⁸ This is an acknowledged simplification considering the large variety and lack of data on such preparations.

The numerical solution to the model equations forms the in silico simulation tool. With the set of 40 virtual patient S_I(t) profiles, any meal or subcutaneous insulin input and its effect on glycemia can be simulated with the assumption that S_I is independent of the inputs administered, i.e., the virtual patient. This opens the possibility of simulating any glycemic control protocol, even current clinical methods.^55–57 An initial validation would be to replicate long-term glycemic control outcomes, e.g., hemoglobin A1c.

Results

To gauge the model fit to data, absolute and absolute percentage errors of the G(t) model fit to the AIDA on-line² patient data cohort are shown in Tables 3 and 4. In Table 3, per patient errors are shown, whereas total errors over the entire cohort are shown in Table 4. A sample G(t) fit is also shown in Figure 8.

Table 3.

Per Patient Absolute and Absolute Percentage G(t) Model Fit Errors to Patient Cohort Data (n = 40) from AIDA On-line²

	Absolute % G(t) fit error			Absolute G(t) fit error (mmol/liter)
Case number	Median	5th percentile	95th percentile	Median	5th percentile	95th percentile
1	0.84	0.08	2.87	0.10	0.01	0.43
2	1.28	0.05	7.14	0.10	0.00	0.50
3	1.19	0.02	3.56	0.12	0.00	0.37
4	1.72	0.05	7.84	0.12	0.00	0.42
5	0.94	0.11	3.24	0.11	0.01	0.36
6	1.25	0.04	4.89	0.13	0.00	0.53
7	1.93	0.11	4.52	0.07	0.00	0.20
8	0.96	0.05	4.46	0.09	0.00	0.49
9	1.03	0.10	2.94	0.12	0.01	0.39
10	1.92	0.28	4.27	0.17	0.02	0.39
11	3.09	0.20	14.04	0.20	0.01	2.39
12	1.23	0.05	8.78	0.13	0.00	0.70
13	1.08	0.09	4.80	0.07	0.00	0.26
14	0.47	0.04	3.43	0.05	0.00	0.47
15	1.00	0.12	3.84	0.08	0.01	0.29
16	1.61	0.11	5.84	0.18	0.01	0.59
17	1.31	0.06	4.38	0.13	0.00	0.38
18	3.34	0.32	7.84	0.21	0.03	0.80
19	1.23	0.09	4.14	0.16	0.01	0.52
20	2.89	0.17	21.39	0.13	0.00	0.89
21	2.73	0.49	7.17	0.32	0.06	0.93
22	0.88	0.11	3.95	0.09	0.01	0.50
23	1.50	0.03	4.96	0.11	0.00	0.33
24	1.98	0.21	6.67	0.14	0.01	0.43
25	1.37	0.02	7.44	0.09	0.00	0.59
26	0.67	0.09	2.48	0.07	0.01	0.31
27	0.69	0.08	2.74	0.06	0.01	0.27
28	0.81	0.09	6.27	0.05	0.00	0.36
29	0.81	0.15	4.48	0.07	0.01	0.40
30	9.36	0.98	37.10	0.72	0.06	2.56
31	1.16	0.11	8.67	0.08	0.01	0.44
32	0.60	0.06	2.36	0.06	0.01	0.28
33	0.93	0.13	2.72	0.10	0.01	0.29
34	1.84	0.01	5.93	0.15	0.00	0.44
35	3.30	0.18	11.70	0.26	0.01	1.06
36	1.36	0.19	5.05	0.12	0.01	0.39
37	0.94	0.03	3.38	0.12	0.00	0.46
38	0.73	0.08	3.07	0.08	0.01	0.40
39	1.29	0.04	5.53	0.11	0.00	0.51
40	1.70	0.20	7.20	0.10	0.01	0.31
Median	1.24	0.09	4.85	0.11	0.01	0.43
Range	0.47–9.36	0.01–0.98	2.36–37.10	0.05–0.72	0.00–0.06	0.20–2.56

Table 4.

Total Absolute and Absolute Percentage G(t) Model Fit Errors to Patient Cohort Data (n = 40) from AIDA On-line²

Absolute % G(t) fit error			Absolute G(t) fit error (mmol/liter)
Median	5th percentile	95th percentile	Median	5th percentile	95th percentile
1.33	0.08	7.20	0.12	0.01	0.56

Figure 8.

G(t) model fit to glucose measurement data for Patient 1 shown with glucose measurement data from AIDA on-line².

From Table 3, the per patient median (95% range) absolute percentage error in G(t) is 1.24% (0.09–4.85%), which translates into a per patient absolute error in G(t) of 0.11 mmol/liter (0.01–0.43 mmol/liter). Over the entire cohort the figures are 1.33% (0.08–7.20%) and 0.12 mmol/liter (0.01–0.56 mmol/liter), which are similar. The errors reported are extremely low and within the measurement errors of clinical methods of glucose measurement in current use. This shows that the model and S_I identification method is capable of capturing all patient G(t) dynamics, which will produce a more physiologically accurate simulation.

Conclusions

An in silico simulation tool was presented that utilizes an extended model of glucose kinetics, a simple glucose rate of appearance model, and the novel application of a subcutaneous insulin pharmacokinetic model. Models are identified to a physiological cohort of type 1 diabetes virtual patients. To corroborate the approach initially, an in silico simulation with data from the patient cohort using the virtual patient simulation method is planned.

Abbreviations

CGM

continuous glucose measurement

CSII

continuous subcutaneous insulin infusion

HGP

hepatic glucose production

IDDM

insulin-dependent diabetes mellitus

MDI

multiple daily injection

SMBG

self-monitoring blood glucose

TBGU

total body glucose uptake